Finite Group Databasebeta

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#72341383

order1536
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC4×B(4,𝔽2) ⋊ D3

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, hypocenter#997C22×C4
FrattiniC24C22×S4
FittingC4×B(4,𝔽2)D3
prime radicalC2×#73026C2
cosocleC2×#997C23
hypercenterC2×C4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C22 ◁ C2×C4

1336 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 20 points:

(1,3,4,2)(5,13,16,9,7,8,18,14)(6,11,19,17,10,15,20,12);(1,2,4,3)(5,6,7,10)(8,17,16,11,9,19,13,12,18,15,14,20);(8,13)(9,14)(11,15)(12,17)(16,18)(19,20)

Polycyclic presentation:

2^4,3,2^5|b^2=d,c^2=dj,f^2=j,g^2=j,c^b=cj,e^c=e^2,f^a=fij,f^b=fij,f^c=gj,f^e=g,g^a=ghj,g^b=ghj,g^c=fj,g^e=fgj,g^f=gj,h^c=i,h^e=hi,h^f=hj,i^c=h,i^e=h,i^g=ij

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341385, C4×#11714171, #72341402, …
maximal normal subgroup of#72341385, C4×#11714171, #72341402, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);